Subgroups — C₈₉×#29212 #36407060

Order 22784.

710 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈₉×#2859	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊Q₈×C₈₉	C₂	1	1	yes	maximal, characteristic
3	C₈₉×#2059	C₂	1	4	yes	maximal
4	C₈₉×#1993	C₂	1	2	yes	maximal
5	C₈₉×#1948	C₂	1	2	yes	maximal
6	C₈₉×#1947	C₂	1	2	yes	maximal
7	C₈₉×#1847	C₂	1	2	yes	maximal
8	C₈₉×#1767	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
10	C₂×#125×C₈₉	C₂²	1	2	yes	—
11	C₂×C₄×Q₈×C₈₉	—	2	1	no	—
12	C₂×#107×C₈₉	C₂²	1	4	yes	—
13	C₂×#106×C₈₉	C₂²	1	4	yes	—
14	C₂×C₄⋊C₈×C₈₉	C₂²	1	2	yes	—
15	C₂×#97×C₈₉	C₂²	1	2	yes	—
16	#391×C₈₉	—	2	1	no	—
17	#389×C₈₉	—	2	1	no	—
18	#389×C₈₉	—	2	1	no	—
19	#389×C₈₉	—	2	1	no	—
20	#389×C₈₉	—	2	1	no	—
21	#389×C₈₉	—	2	1	no	—
22	#389×C₈₉	C₂²	1	2	yes	—
23	#384×C₈₉	C₂²	1	1	yes	characteristic
24	#384×C₈₉	C₂²	1	1	yes	characteristic
25	#384×C₈₉	C₂²	1	1	yes	characteristic
26	#382×C₈₉	C₂²	1	2	yes	—
27	C₄×C₄⋊C₄×C₈₉	—	2	1	no	—
28	C₄×C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
29	C₄³×C₈₉	C₂²	1	1	yes	characteristic
30	#340×C₈₉	C₂²	1	4	yes	—
31	#340×C₈₉	C₂²	1	4	yes	—
32	#336×C₈₉	C₂²	1	2	yes	—
33	C₂²×Q₈×C₈₉	—	2	1	no	—
34	C₂²×C₈×C₈₉	C₂³	1	4	yes	—
35	C₄⋊Q₈×C₈₉	—	2	2	no	—
36	C₄⋊Q₈×C₈₉	Q₈	1	4	yes	—
37	#125×C₈₉	—	4	4	no	—
38	C₄×Q₈×C₈₉	—	4	4	no	—
39	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
40	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
41	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
42	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
43	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
44	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
45	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
46	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
47	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
48	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
49	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
50	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
51	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
52	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
53	C₂×C₄²×C₈₉	—	2	2	no	—
54	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
55	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
56	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
57	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
58	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
59	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
60	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
61	#107×C₈₉	—	4	4	no	—
62	#106×C₈₉	—	4	4	no	—
63	C₄⋊C₈×C₈₉	Q₈	1	8	yes	complemented
64	#97×C₈₉	—	2	4	no	—
65	#95×C₈₉	—	2	2	no	—
66	#95×C₈₉	—	2	2	no	—
67	#95×C₈₉	—	2	2	no	—
68	#95×C₈₉	—	2	2	no	—
69	C₂×Q₈×C₈₉	—	2	4	no	—
70	C₂×Q₈×C₈₉	—	4	2	no	—
71	C₂×Q₈×C₈₉	—	4	2	no	—
72	C₂²×C₄×C₈₉	—	2	2	no	—
73	C₂²×C₄×C₈₉	—	2	2	no	—
74	C₂²×C₄×C₈₉	—	2	2	no	—
75	C₂²×C₄×C₈₉	—	2	2	no	—
76	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
83	C₂×C₈×C₈₉	—	2	4	no	—
84	C₂×C₈×C₈₉	—	2	4	no	—
85	C₂×C₈×C₈₉	C₂×Q₈	1	8	yes	—
86	C₄⋊C₄×C₈₉	—	2	4	no	—
87	C₄⋊C₄×C₈₉	—	2	2	no	—
88	C₄⋊C₄×C₈₉	—	2	2	no	—
89	C₄⋊C₄×C₈₉	—	2	4	no	—
90	C₄⋊C₄×C₈₉	—	2	2	no	—
91	C₄⋊C₄×C₈₉	—	2	4	no	—
92	C₄⋊C₄×C₈₉	—	4	2	no	—
93	C₄⋊C₄×C₈₉	—	4	2	no	—
94	C₄⋊C₄×C₈₉	—	4	2	no	—
95	C₄⋊C₄×C₈₉	—	4	2	no	—
96	C₄⋊C₄×C₈₉	—	4	4	no	—
97	C₄⋊C₄×C₈₉	—	4	4	no	—
98	C₄⋊C₄×C₈₉	—	4	4	no	—
99	C₄⋊C₄×C₈₉	—	4	2	no	—
100	C₄⋊C₄×C₈₉	—	4	2	no	—
101	C₄⋊C₄×C₈₉	C₂×Q₈	1	4	yes	—
102	C₄²×C₈₉	—	2	2	no	—
103	C₄²×C₈₉	—	2	2	no	—
104	C₄²×C₈₉	—	2	2	no	—
105	C₄²×C₈₉	—	4	1	no	—
106	C₄²×C₈₉	—	4	1	no	—
107	C₄²×C₈₉	—	4	4	no	—
108	C₄²×C₈₉	C₂×Q₈	1	2	yes	—
109	C₄²×C₈₉	C₂×Q₈	1	4	yes	—
110	C₄²×C₈₉	#41	1	2	yes	—
111	C₂³×C₈₉	C₂²×D₄	1	1	yes	characteristic
112	Q₈×C₈₉	—	4	8	no	—
113	C₂×C₄×C₈₉	—	2	1	no	—
114	C₂×C₄×C₈₉	—	2	1	no	—
115	C₂×C₄×C₈₉	—	2	1	no	—
116	C₂×C₄×C₈₉	—	2	1	no	—
117	C₂×C₄×C₈₉	—	2	1	no	—
118	C₂×C₄×C₈₉	—	2	1	no	—
119	C₂×C₄×C₈₉	—	2	4	no	—
120	C₂×C₄×C₈₉	—	2	1	no	—
121	C₂×C₄×C₈₉	—	2	1	no	—
122	C₂×C₄×C₈₉	—	2	1	no	—
123	C₂×C₄×C₈₉	—	2	1	no	—
124	C₂×C₄×C₈₉	—	2	1	no	—
125	C₂×C₄×C₈₉	—	2	1	no	—
126	C₂×C₄×C₈₉	—	4	2	no	—
127	C₂×C₄×C₈₉	—	4	2	no	—
128	C₂×C₄×C₈₉	—	4	2	no	—
129	C₂×C₄×C₈₉	—	4	2	no	—
130	C₂×C₄×C₈₉	—	4	2	no	—
131	C₂×C₄×C₈₉	—	4	2	no	—
132	C₂×C₄×C₈₉	—	4	2	no	—
133	C₂×C₄×C₈₉	—	4	2	no	—
134	C₂×C₄×C₈₉	—	4	2	no	—
135	C₂×C₄×C₈₉	—	4	2	no	—
136	C₂×C₄×C₈₉	—	4	2	no	—
137	C₂×C₄×C₈₉	C₂²⋊Q₈	1	2	yes	—
138	C₂×C₄×C₈₉	C₂²⋊Q₈	1	2	yes	—
139	C₂×C₄×C₈₉	C₄⋊Q₈	1	2	yes	—
140	C₂×C₄×C₈₉	C₄⋊Q₈	1	2	yes	—
141	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	2	yes	—
142	C₂×C₄×C₈₉	#137	1	2	yes	—
143	C₂×C₄×C₈₉	C₂²×Q₈	1	2	yes	—
144	C₂×C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
145	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
146	C₂×C₄×C₈₉	H(5,𝔽₂)	1	2	yes	—
147	C₈×C₈₉	—	2	8	no	—
148	C₂²×C₈₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
149	C₂²×C₈₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
150	C₂²×C₈₉	#535	1	1	yes	characteristic
151	C₂²×C₈₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
152	C₂²×C₈₉	C₂×#137	1	1	yes	characteristic
153	C₂²×C₈₉	#575	1	1	yes	characteristic
154	C₂²×C₈₉	#575	1	1	yes	characteristic
155	C₄×C₈₉	—	2	2	no	—
156	C₄×C₈₉	—	2	2	no	—
157	C₄×C₈₉	—	2	2	no	—
158	C₄×C₈₉	—	4	4	no	—
159	C₄×C₈₉	—	4	1	no	—
160	C₄×C₈₉	—	4	1	no	—
161	C₄×C₈₉	—	8	2	no	—
162	C₄×C₈₉	—	8	2	no	—
163	C₄×C₈₉	—	8	2	no	—
164	C₄×C₈₉	#501	1	4	yes	—
165	C₄×C₈₉	#554	1	2	yes	—
166	C₄×C₈₉	Q₈²	1	2	yes	—
167	#29212	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
168	C₂×C₈₉	#2567	1	1	yes	characteristic
169	C₂×C₈₉	#3077	1	1	yes	characteristic
170	C₂×C₈₉	#3078	1	1	yes	characteristic
171	C₂×C₈₉	C₂×#501	1	1	yes	characteristic
172	C₂×C₈₉	#3166	1	1	yes	characteristic
173	C₂×C₈₉	#3214	1	1	yes	characteristic
174	C₂×C₈₉	#3216	1	1	yes	characteristic
175	#2859	C₂×C₈₉	1	1	yes	characteristic
176	C₄×C₄⋊Q₈	C₂×C₈₉	1	1	yes	characteristic
177	#2059	C₂×C₈₉	1	4	yes	—
178	#1993	C₂×C₈₉	1	2	yes	—
179	#1948	C₂×C₈₉	1	2	yes	—
180	#1947	C₂×C₈₉	1	2	yes	—
181	#1847	C₂×C₈₉	1	2	yes	—
182	#1767	C₂×C₈₉	1	1	yes	characteristic
183	C₈₉	#29212	1	1	yes	characteristic, direct product factor, Sylow
184	C₂×C₄⋊Q₈	C₂²×C₈₉	1	1	yes	characteristic
185	C₂×#125	C₂²×C₈₉	1	2	yes	—
186	C₂×C₄×Q₈	—	2	1	no	—
187	C₂×#107	C₂²×C₈₉	1	4	yes	—
188	C₂×#106	C₂²×C₈₉	1	4	yes	—
189	C₂×C₄⋊C₈	C₂²×C₈₉	1	2	yes	—
190	C₂×#97	C₂²×C₈₉	1	2	yes	—
191	#391	—	2	1	no	—
192	#389	—	2	1	no	—
193	#389	—	2	1	no	—
194	#389	—	2	1	no	—
195	#389	—	2	1	no	—
196	#389	—	2	1	no	—
197	#389	C₂²×C₈₉	1	2	yes	—
198	#384	C₂²×C₈₉	1	1	yes	characteristic
199	#384	C₂²×C₈₉	1	1	yes	characteristic
200	#384	C₂²×C₈₉	1	1	yes	characteristic
201	#382	C₂²×C₈₉	1	2	yes	—
202	C₄×C₄⋊C₄	—	2	1	no	—
203	C₄×C₄⋊C₄	C₂²×C₈₉	1	2	yes	—
204	C₄³	C₂²×C₈₉	1	1	yes	characteristic
205	#340	C₂²×C₈₉	1	4	yes	—
206	#340	C₂²×C₈₉	1	4	yes	—
207	#336	C₂²×C₈₉	1	2	yes	—
208	C₂²×Q₈	—	2	1	no	—
209	C₂²×C₈	C₂³×C₈₉	1	4	yes	—
210	C₄⋊Q₈	—	2	2	no	—
211	C₄⋊Q₈	Q₈×C₈₉	1	4	yes	—
212	#125	—	4	4	no	—
213	C₄×Q₈	—	4	4	no	—
214	C₂×C₄⋊C₄	—	2	1	no	—
215	C₂×C₄⋊C₄	—	2	1	no	—
216	C₂×C₄⋊C₄	—	2	2	no	—
217	C₂×C₄⋊C₄	—	2	2	no	—
218	C₂×C₄⋊C₄	—	2	2	no	—
219	C₂×C₄⋊C₄	—	2	1	no	—
220	C₂×C₄⋊C₄	—	2	1	no	—
221	C₂×C₄⋊C₄	—	2	1	no	—
222	C₂×C₄⋊C₄	—	2	1	no	—
223	C₂×C₄⋊C₄	—	2	1	no	—
224	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
225	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
226	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
227	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
228	C₂×C₄²	—	2	2	no	—
229	C₂×C₄²	D₄×C₈₉	1	1	yes	characteristic
230	C₂×C₄²	D₄×C₈₉	1	1	yes	characteristic
231	C₂×C₄²	D₄×C₈₉	1	1	yes	characteristic
232	C₂×C₄²	D₄×C₈₉	1	1	yes	characteristic
233	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
234	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
235	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
236	#107	—	4	4	no	—
237	#106	—	4	4	no	—
238	C₄⋊C₈	Q₈×C₈₉	1	8	yes	complemented
239	#97	—	2	4	no	—
240	#95	—	2	2	no	—
241	#95	—	2	2	no	—
242	#95	—	2	2	no	—
243	#95	—	2	2	no	—
244	C₂×Q₈	—	2	4	no	—
245	C₂×Q₈	—	4	2	no	—
246	C₂×Q₈	—	4	2	no	—
247	C₂²×C₄	—	2	2	no	—
248	C₂²×C₄	—	2	2	no	—
249	C₂²×C₄	—	2	2	no	—
250	C₂²×C₄	—	2	2	no	—
251	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
252	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
253	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
254	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
255	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
256	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
257	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
258	C₂×C₈	—	2	4	no	—
259	C₂×C₈	—	2	4	no	—
260	C₂×C₈	C₂×Q₈×C₈₉	1	8	yes	—
261	C₄⋊C₄	—	2	4	no	—
262	C₄⋊C₄	—	2	2	no	—
263	C₄⋊C₄	—	2	2	no	—
264	C₄⋊C₄	—	2	4	no	—
265	C₄⋊C₄	—	2	2	no	—
266	C₄⋊C₄	—	2	4	no	—
267	C₄⋊C₄	—	4	2	no	—
268	C₄⋊C₄	—	4	4	no	—
269	C₄⋊C₄	—	4	4	no	—
270	C₄⋊C₄	—	4	4	no	—
271	C₄⋊C₄	—	4	2	no	—
272	C₄⋊C₄	—	4	2	no	—
273	C₄⋊C₄	—	4	2	no	—
274	C₄⋊C₄	—	4	2	no	—
275	C₄⋊C₄	—	4	2	no	—
276	C₄⋊C₄	C₂×Q₈×C₈₉	1	4	yes	—
277	C₄²	—	2	2	no	—
278	C₄²	—	2	2	no	—
279	C₄²	—	2	2	no	—
280	C₄²	—	4	1	no	—
281	C₄²	—	4	1	no	—
282	C₄²	—	4	4	no	—
283	C₄²	C₂×Q₈×C₈₉	1	2	yes	—
284	C₄²	C₂×Q₈×C₈₉	1	4	yes	—
285	C₄²	#41×C₈₉	1	2	yes	—
286	C₂³	C₂²×D₄×C₈₉	1	1	yes	characteristic
287	Q₈	—	4	8	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	4	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	—	4	2	no	—
303	C₂×C₄	—	4	2	no	—
304	C₂×C₄	—	4	2	no	—
305	C₂×C₄	—	4	2	no	—
306	C₂×C₄	—	4	2	no	—
307	C₂×C₄	—	4	2	no	—
308	C₂×C₄	—	4	2	no	—
309	C₂×C₄	—	4	2	no	—
310	C₂×C₄	—	4	2	no	—
311	C₂×C₄	—	4	2	no	—
312	C₂×C₄	C₂²⋊Q₈×C₈₉	1	2	yes	—
313	C₂×C₄	C₂²⋊Q₈×C₈₉	1	2	yes	—
314	C₂×C₄	C₄⋊Q₈×C₈₉	1	2	yes	—
315	C₂×C₄	C₄⋊Q₈×C₈₉	1	2	yes	—
316	C₂×C₄	AGL(1,ℤ/8)×C₈₉	1	2	yes	—
317	C₂×C₄	#137×C₈₉	1	2	yes	—
318	C₂×C₄	C₂²×Q₈×C₈₉	1	2	yes	—
319	C₂×C₄	C₂²×Q₈×C₈₉	1	1	yes	characteristic
320	C₂×C₄	C₂×#41×C₈₉	1	1	yes	characteristic
321	C₂×C₄	H(5,𝔽₂)×C₈₉	1	2	yes	—
322	C₈	—	2	8	no	—
323	C₂²	C₂×C₂²⋊Q₈×C₈₉	1	1	yes	characteristic
324	C₂²	C₂×C₄⋊Q₈×C₈₉	1	1	yes	characteristic
325	C₂²	#535×C₈₉	1	1	yes	characteristic
326	C₂²	C₂×AGL(1,ℤ/8)×C₈₉	1	1	yes	characteristic
327	C₂²	C₂×#137×C₈₉	1	1	yes	characteristic
328	C₂²	#575×C₈₉	1	1	yes	characteristic
329	C₂²	#575×C₈₉	1	1	yes	characteristic
330	C₄	—	2	2	no	—
331	C₄	—	2	2	no	—
332	C₄	—	2	2	no	—
333	C₄	—	4	4	no	—
334	C₄	—	4	1	no	—
335	C₄	—	4	1	no	—
336	C₄	—	8	2	no	—
337	C₄	—	8	2	no	—
338	C₄	—	8	2	no	—
339	C₄	#501×C₈₉	1	4	yes	—
340	C₄	#554×C₈₉	1	2	yes	—
341	C₄	Q₈²×C₈₉	1	2	yes	—
342	C₂	C₈₉×#2567	1	1	yes	characteristic
343	C₂	C₈₉×#3077	1	1	yes	characteristic
344	C₂	C₈₉×#3078	1	1	yes	characteristic
345	C₂	C₂×#501×C₈₉	1	1	yes	characteristic
346	C₂	C₈₉×#3166	1	1	yes	characteristic
347	C₂	C₈₉×#3214	1	1	yes	characteristic
348	C₂	C₈₉×#3216	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₈₉	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₈₉
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₈₉ ◁ C₄³×C₈₉ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
63. C₄⋊C₈×C₈₉	Q₈	semidirect
167. #29212	C₈₉	direct (N × H)
183. C₈₉	#29212	direct (N × H)
238. C₄⋊C₈	Q₈×C₈₉	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₈₉×#2859	2
C₄×C₄⋊Q₈×C₈₉	2
C₈₉×#2059	2
C₈₉×#1993	2
C₈₉×#1948	2
C₈₉×#1947	2
C₈₉×#1847	2
C₈₉×#1767	2
#29212	89

Subgroups — C89×#29212 #36407060

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#29212 #36407060